We can download files about some special planar graphs at the following website, but their files are suffixed with pc.

I don't know how to read them. When I tried to open the file, the contents were garbled.

enter image description here

It feels like they're binaries. Mathematica does a great job of reading the Graph6 format. But is there any hope of reading against this format in this web.

Here's what I tried. For eample, I tried to read a 12-vertex 5-regular planar graph. But I don't understand what output means.


enter image description here

  • $\begingroup$ It seems you need a plantri program for that users.cecs.anu.edu.au/~bdm/plantri $\endgroup$
    – yarchik
    Jun 15, 2022 at 10:21
  • $\begingroup$ Thanks. I know the plantri program. But some sepcail planar graphs in above web are not easily to be obtained by plantri, as stated on the web page. So I thought it would be easier to just read it $\endgroup$
    – licheng
    Jun 15, 2022 at 11:09

1 Answer 1


One option is to generate these planar graphs using the plantri program and ask for Graph6 output, which Mathematica can read (and IGraph/M can read faster). However, Graph6 does not encode the combinatorial embedding like the pc format does. The following is a small function that decodes this format to combinatorial embeddings:

decode[data_] :=
 Module[{d = data, head, vc, g},
  (* skip header if it exists *)
  head = ToCharacterCode[">>planar_code<<"];
  If[Take[d, Min[Length[d], Length[head]]] === head,
    d = Drop[d, Length[head]];
  (* split data at zero separators *)
  d = Most /@ Split[d, #1 =!= 0 &];
  First@Last@Reap@While[Length[d] > 0,
      vc = d[[1, 1]]; (* vertex count *)
      (* get as many neighbour lists as the vertex count *)
      {g, d} = TakeDrop[d, vc];
      (* drop vertex count from first neighbour list *)
      g = MapAt[Rest, g, {1}];
      (* build association for combinatorial embedding *)
      Sow@AssociationThread[Range[vc], Reverse /@ g (* reverse from clockwise to counterclockwise *)]

Note that there is no error checking! If you are interested in a robust version, feel free to open a feature request for IGraph/M, and I will consider including it in the next version.

The expected input is the contents of a valid PC file, as a list of bytes (values between 0..255).


data = Import[
  "http://users.cecs.anu.edu.au/%7Ebdm/data/5reg_20-32.pc", "Byte"]


This returns a list of associations, each representing a graph and its combinatorial embedding. The format is the same that IGraph/M uses, which I think you are familiar with: vertices are associated with a list of their counter-clockwise neighbours.

You can use IGAdjacencyGraph to convert this format to a graph or IGEmbeddingToCoordinates to get planar coordinates for them.

In the following example I chose to use a Tutte layout instead, as it is visually more appealing:

IGLayoutTutte@IGAdjacencyGraph[#] & /@ decode[data]

enter image description here

  • $\begingroup$ That's great, although I don't know why your code works, right. but it deserves to be a new function in the next version of IGraph/M. $\endgroup$
    – licheng
    Jun 15, 2022 at 11:16

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